4 problems

Bisimplicial-vertex conjecture. No MNPD graph contains a bisimplicial vertex.

Even-hole-free conjecture. Every even-hole-free graph is perfectly divisible.

Independence-number-three conjecture. Every graph $G$ with $\alpha(G)\leq 3$ is perfectly divisible.

P5-free conjecture. Every $P_5$-free graph is perfectly divisible.